论文标题
多元等渗回归的后收缩和测试
Posterior Contraction and Testing for Multivariate Isotonic Regression
论文作者
论文摘要
我们考虑具有多个预测变量的非参数回归问题和一个加性误差，其中假定回归函数是坐标的非负责。我们提出了一种贝叶斯方法，以推断多变量单调回归函数，获得后部收缩率，并为多元单调性构建普遍一致的贝叶斯测试程序。为了促进后验分析，我们暂时搁置了形状限制，并在块恒定回归函数上赋予高度正态分布的高度。误差项的未知差异要么由边际最大似然估计估计，要么配备了逆伽马之前。然后，由于误差差异，不受限制的块高度是后验，也是正态分布的后验。为了遵守形状限制，我们将样本从无限制的后部投射到多元单调函数的类别上，从而诱导“投射 - 托管分布”，用于进行推理。在$ \ mathbb {l} _1 $ - metric下，我们证明了基于$ n $独立样本的投射 - 验证验，以最佳速率$ n^{ - 1/（2+d）} $围绕真实单调回归函数。然后，我们基于基于多元单调函数类缩小邻居的后率概率来构建用于多元单调性的贝叶斯测试。我们表明该测试是普遍一致的，即贝叶斯测试的水平为零，任何固定替代方案的功率都归功于一个。此外，我们表明，对于平稳的替代功能，只要其与多元单调函数的距离至少是平滑函数的估计误差的顺序，则功率就达到了。
We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make an inference on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we set aside the shape restrictions temporarily, and endow a prior on blockwise constant regression functions with heights independently normally distributed. The unknown variance of the error term is either estimated by the marginal maximum likelihood estimate or is equipped with an inverse-gamma prior. Then the unrestricted block heights are a posteriori also independently normally distributed given the error variance, by conjugacy. To comply with the shape restrictions, we project samples from the unrestricted posterior onto the class of multivariate monotone functions, inducing the "projection-posterior distribution", to be used for making an inference. Under an $\mathbb{L}_1$-metric, we show that the projection-posterior based on $n$ independent samples contracts around the true monotone regression function at the optimal rate $n^{-1/(2+d)}$. Then we construct a Bayesian test for multivariate monotonicity based on the posterior probability of a shrinking neighborhood of the class of multivariate monotone functions. We show that the test is universally consistent, that is, the level of the Bayesian test goes to zero, and the power at any fixed alternative goes to one. Moreover, we show that for a smooth alternative function, power goes to one as long as its distance to the class of multivariate monotone functions is at least of the order of the estimation error for a smooth function.
